Gaussian polytopes with large Banach-Mazur distance to the cross-polytope

Abstract

Let $B_1^n$ be the standard cross-polytope in $\mathbb R^n$, let $g_1,\ldots,g_m$ be independent standard Gaussian vectors in $\mathbb R^n$, and set $G_m=\operatorname{conv}{\pm g_1,\ldots,\pm g_m}$. For $m=n^3$ it is proved that $$ \mathbb P\left\{d_{\mathrm{BM}}(G_m,B_1^n)\geqslant c n^{5/8}(\ln n)^{-5/8}\right\}\geqslant 1-\frac2n $$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n)^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.